21  Relationships Between Variables

21.1 The question

“Does a mother’s age predict the birth weight of her baby?”

All previous chapters studied one variable at a time. Now we ask: do two variables move together? And if so, how strongly?

21.2 Scatter plots

The first tool for any relationship: plot one variable against the other. With 9 000 points, dots overlap completely. Standard fixes:

  1. Jitter — add small random noise so overlapping points spread out
  2. Alpha — make each dot semi-transparent; dense regions appear darker
  3. Bin and mean — divide x into bins, plot mean y per bin (shows trend clearly)
  4. Hexbin — 2D histogram using hexagonal bins

21.3 Characterising relationships

Before computing a correlation coefficient, ask:

  1. Is the relationship linear or curved?
  2. Is it monotone (always increasing or always decreasing)?
  3. Are there subgroups that behave differently?
  4. Are there outliers that dominate the relationship?

A correlation coefficient summarises only point 1. Always look at the scatter plot first.

21.4 Covariance

\operatorname{Cov}(X, Y) \;=\; \frac{1}{n}\sum_{i=1}^n (x_i - \bar{x})(y_i - \bar{y})

  • Positive: when X is high, Y tends to be high
  • Negative: when X is high, Y tends to be low

Covariance has units (lbs × years for our example) — comparing covariances across pairs is meaningless.

21.5 Pearson’s correlation

Normalise covariance by the product of standard deviations:

r \;=\; \frac{\operatorname{Cov}(X, Y)}{\sigma_X \sigma_Y} \;=\; \frac{\sum(x_i - \bar{x})(y_i - \bar{y})} {\sqrt{\sum(x_i - \bar{x})^2 \sum(y_i - \bar{y})^2}}

Now r \in [-1, 1]:

  • r = +1: perfect positive linear relationship
  • r = -1: perfect negative linear relationship
  • r = 0: no linear relationship
Warningr = 0 does not mean “no relationship”

It means no linear relationship. The data can have a strong curved relationship and still produce r = 0. Always plot first.

21.6 Spearman’s rank correlation

Pearson’s r applied to ranks, not values:

  1. Replace each x_i with its rank
  2. Replace each y_i with its rank
  3. Compute Pearson’s r on the ranks

\rho \;=\; 1 - \frac{6 \sum d_i^2}{n(n^2 - 1)}

where d_i is the difference in ranks.

Use Spearman when: the relationship is monotone but not linear; outliers are present; data is ordinal.

21.7 Computing it on NSFG

import sys, os
import numpy as np
import matplotlib.pyplot as plt

sys.path.insert(0, os.path.dirname(os.path.abspath("__file__")) or ".")
from _nsfg import load_groups, COLORS

live, _, _ = load_groups()

df = live[["agepreg", "totalwgt_lb", "prglngth"]].dropna()
df = df[(df["totalwgt_lb"] > 0) & (df["totalwgt_lb"] < 20)]
df = df[(df["agepreg"] > 10) & (df["agepreg"] < 50)]

age    = df["agepreg"].values
weight = df["totalwgt_lb"].values
prg    = df["prglngth"].values


def covariance(x, y):
    return np.mean((x - x.mean()) * (y - y.mean()))


def pearson_r(x, y):
    return covariance(x, y) / (x.std() * y.std())


def rank_array(x):
    order = np.argsort(x)
    ranks = np.empty(len(x))
    ranks[order] = np.arange(1, len(x) + 1)
    return ranks


def spearman_rho(x, y):
    return pearson_r(rank_array(x), rank_array(y))


print("── Mother's age vs birth weight ─────────────────────")
print(f"  Covariance : {covariance(age, weight):.4f} (lbs × years)")
print(f"  Pearson r  : {pearson_r(age, weight):+.4f}")
print(f"  Spearman ρ : {spearman_rho(age, weight):+.4f}")
print()
print("── Pregnancy length vs birth weight ─────────────────")
print(f"  Pearson r  : {pearson_r(prg, weight):+.4f}")
print(f"  Spearman ρ : {spearman_rho(prg, weight):+.4f}")
── Mother's age vs birth weight ─────────────────────
  Covariance : 0.5561 (lbs × years)
  Pearson r  : +0.0684
  Spearman ρ : +0.0940

── Pregnancy length vs birth weight ─────────────────
  Pearson r  : +0.4729
  Spearman ρ : +0.3335

Pregnancy-length vs weight: strong positive correlation — longer pregnancies produce heavier babies. Age vs weight: weak positive correlation — but Pearson’s r misses something the binned-mean plot will reveal.

21.8 Binned mean — revealing nonlinearity

age_bins = np.arange(10, 50, 5)
bin_means, bin_centers, bin_counts = [], [], []
for lo, hi in zip(age_bins[:-1], age_bins[1:]):
    mask = (age >= lo) & (age < hi)
    if mask.sum() > 10:
        bin_means.append(weight[mask].mean())
        bin_centers.append((lo + hi) / 2)
        bin_counts.append(int(mask.sum()))

print(f"  {'Age range':>14}  {'Mean wgt (lbs)':>15}  {'n':>6}")
for c, m, n in zip(bin_centers, bin_means, bin_counts):
    print(f"  {c-2.5:.0f}–{c+2.5:.0f} years   {m:>12.3f}    {n:>6,}")
       Age range   Mean wgt (lbs)       n
  10–15 years          7.331        61
  15–20 years          7.093     1,860
  20–25 years          7.282     2,975
  25–30 years          7.387     2,346
  30–35 years          7.395     1,399
  35–40 years          7.421       406
  40–45 years          7.128        37

The relationship is U-shaped: very young mothers (teens) have lighter babies; mothers in their 20s and 30s have the heaviest; mothers above 40 have slightly lighter babies again. Pearson’s r flattens this U into a single near-zero number.

21.9 Correlation and causation

The most abused concept in statistics.

Correlation tells you two variables are associated. It says nothing about why. Ice cream sales and drowning rates are positively correlated. The cause is summer (a confound), not ice cream.

Establishing causation requires either:

  1. A randomised controlled experiment
  2. A natural experiment (quasi-random assignment)
  3. Careful matching/adjustment for confounds

Observational data like NSFG can only establish association.

21.10 Visualising it

np.random.seed(42)

fig, axes = plt.subplots(2, 2, figsize=(12, 9))

# 1. Raw scatter — too dense
ax = axes[0, 0]
ax.scatter(age, weight, alpha=0.05, s=5, color=COLORS["first"])
ax.set_xlabel("Mother's age (years)")
ax.set_ylabel("Birth weight (lbs)")
ax.set_title("Scatter (alpha=0.05)")

# 2. Jitter + binned mean
ax = axes[0, 1]
jitter_age = age + np.random.normal(0, 0.3, len(age))
ax.scatter(jitter_age, weight, alpha=0.05, s=5, color=COLORS["first"])
ax.plot(bin_centers, bin_means, color=COLORS["highlight"],
        linewidth=2.5, marker="o", markersize=6, label="Binned mean")
ax.set_xlabel("Mother's age (years)")
ax.set_ylabel("Birth weight (lbs)")
ax.set_title("Jittered Scatter + Binned Mean")
ax.legend()

# 3. Binned mean alone — U shape clearer
ax = axes[1, 0]
ax.plot(bin_centers, bin_means, color=COLORS["highlight"],
        linewidth=2.5, marker="o", markersize=8)
ax.fill_between(bin_centers, bin_means, alpha=0.15, color=COLORS["highlight"])
ax.set_xlabel("Mother's age (years)")
ax.set_ylabel("Mean birth weight (lbs)")
ax.set_title("Nonlinear: Age vs Birth Weight")
ax.set_ylim(6.5, 8.5)

# 4. Pearson vs Spearman across pairs
ax = axes[1, 1]
pairs = ["Age vs\nWeight", "Prg Length vs\nWeight"]
pearson_vals  = [pearson_r(age, weight),    pearson_r(prg, weight)]
spearman_vals = [spearman_rho(age, weight), spearman_rho(prg, weight)]
x = np.arange(len(pairs))
ax.bar(x - 0.2, pearson_vals,  width=0.4, color=COLORS["other"],
       alpha=0.85, label="Pearson r")
ax.bar(x + 0.2, spearman_vals, width=0.4, color="#FF9800",
       alpha=0.85, label="Spearman ρ")
ax.axhline(0, color="black", linewidth=0.8)
ax.set_xticks(x)
ax.set_xticklabels(pairs, fontsize=9)
ax.set_ylabel("Correlation")
ax.set_title("Pearson vs Spearman")
ax.legend()
ax.set_ylim(-0.1, 0.5)

plt.tight_layout()
plt.show()

Raw scatter, jittered scatter with binned mean, the binned-mean plot, and Pearson vs Spearman.

21.11 Exercises

  1. Make a scatter plot of mother’s age vs birth weight with alpha=0.1. What pattern do you see?
  2. Compute Pearson’s r between age and birth weight. Is it strong?
  3. Compute Spearman’s \rho for the same pair. How does it compare?
  4. Bin mother’s age into 5-year groups and plot mean birth weight per bin. What shape do you see?
  5. Compute r between pregnancy length and birth weight. Which pair is more strongly correlated?

21.12 Glossary

scatter plot — one point per observation.

jitter — small random noise to avoid overplotting.

covariance — \operatorname{Cov}(X,Y); direction of linear association; has units.

Pearson’s r — normalised covariance; unit-free; linear association.

Spearman’s ρ — Pearson’s r on ranks; monotone association; robust to outliers.

confound — a third variable that causes both X and Y.

correlation ≠ causation — association does not imply causation.